Lets say I started to dig a 10 foot wide hole from my backyard to the exact opposite side of the earth. What would happen to gravity when I reached the centre of the earth and started to dig up (or would that still be down)?? If I made it all the way through, how would that be physically possible?

Yea, I think force due to gravity is inversly proportional to the radius of the earth squared. So the further down you go (ignoring effects of heat), you would feel less and less in weight. Once you get to the center, if the earth where the only object in the universe, you would probably just float because you would be at the center of gravity and mass of the entire system.

You would weigh more all the way down to the center, but, you would stop because the gravity is equal all the way around the circumfrence of the earth... There would be an equal force each way... You may crush....

What you'd find along the way: Bill Buckner's firstbaseman's mitt, chads from the Florida ballots, the missing 727 from Africa, all of JCS's unsigned report cards from elementary school, and the end of the uploaded a.net picture queue.

Provided Earth would be solid and a perfect sphere, gravitational force would decrease with increasing depth because the forces of all the layers above you would cancel each other out and onkly the amount of mass between you and the center would attract you. At the center you would be weightless.

An object thrown into a borehole right through the planet would (provided no friction) oscillate through the center.